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A proportion is an equation that shows two ratios or fractions are equivalent. When solving proportions, our main goal is to find the missing value that makes this equality true. Proportions are often represented as two fractions set equal to each other, like \( \frac{a}{b} = \frac{c}{d} \). In our given problem, we have a proportion \( \frac{2w-1}{7} = \frac{3-6w}{4} \). Here, we need to determine the value of \( w \) that makes both sides of the equation equal. Solving proportions is an essential part of algebra because it helps us understand the relationship between quantities, allowing for predictions and conclusions about various scenarios. With some straightforward mathematical steps, we can easily find the value we're looking for.

Cross-multiplication is a technique used to solve proportions. It is based on the idea that if two fractions are equal, then their cross-products are also equal. When you have a proportion \( \frac{a}{b} = \frac{c}{d} \), you can apply cross-multiplication, resulting in the equation \( ad = bc \).

In the exercise, we apply cross-multiplication by multiplying the numerator of each fraction by the denominator of the other fraction, which gives us:

• \( (2w - 1) \times 4 = (3 - 6w) \times 7 \)

This step effectively eliminates the fractions and gives us a simple equation to solve. Cross-multiplication works reliably for any known proportion, making it a powerful tool in solving algebraic equations.

After cross-multiplication, the next step is to use distribution. Distribution involves multiplying each term inside a parenthesis by the factor outside the parenthesis. This helps break down expressions into simpler forms.

In our example, after cross-multiplying, we have

• \((2w - 1) \, (\times 4) = 8w - 4\)
• \((3 - 6w) \, (\times 7) = 21 - 42w\)

By applying the distributive property, which states \(a(b+c) = ab + ac\), each term gets expanded individually. This step is crucial because it simplifies the equation and prepares it for solving. Distribution is a versatile technique that makes handling multi-term algebraic expressions far more manageable.

After distributing, we have a linear equation which is an equation where the highest power of the variable is 1. In solving the equation, \(8w - 4 = 21 - 42w\), our goal is to isolate the variable \(w\). We do this by moving all terms containing \(w\) to one side and constant terms to the other side, using addition or subtraction.

First, add \(42w\) to both sides:

• \(8w + 42w - 4 = 21\)

Simplify by combining like terms, \(50w - 4 = 21\). Then, add 4 to both sides to get \(50w = 25\). Finally, divide both sides by 50 to solve for \(w\), leading to \(w = \frac{25}{50} = \frac{1}{2}\). Solving linear equations involves organizing terms and performing basic arithmetic operations. The simplicity of linear equations makes them foundational for understanding more complex algebraic concepts.